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Properties

Label 2-1408-176.43-c1-0-22
Degree $2$
Conductor $1408$
Sign $-0.644 + 0.764i$
Analytic cond. $11.2429$
Root an. cond. $3.35304$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.21 − 2.21i)3-s + (0.858 + 0.858i)5-s − 1.49i·7-s + 6.85i·9-s + (3.03 + 1.33i)11-s + (−2.98 − 2.98i)13-s − 3.80i·15-s + 2.42i·17-s + (3.38 − 3.38i)19-s + (−3.31 + 3.31i)21-s + 0.410·23-s − 3.52i·25-s + (8.55 − 8.55i)27-s + (6.31 + 6.31i)29-s − 8.16i·31-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)3-s + (0.383 + 0.383i)5-s − 0.564i·7-s + 2.28i·9-s + (0.915 + 0.401i)11-s + (−0.827 − 0.827i)13-s − 0.983i·15-s + 0.588i·17-s + (0.775 − 0.775i)19-s + (−0.723 + 0.723i)21-s + 0.0855·23-s − 0.705i·25-s + (1.64 − 1.64i)27-s + (1.17 + 1.17i)29-s − 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1408\)    =    \(2^{7} \cdot 11\)
Sign: $-0.644 + 0.764i$
Analytic conductor: \(11.2429\)
Root analytic conductor: \(3.35304\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1408} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1408,\ (\ :1/2),\ -0.644 + 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9620675533\)
\(L(\frac12)\) \(\approx\) \(0.9620675533\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-3.03 - 1.33i)T \)
good3 \( 1 + (2.21 + 2.21i)T + 3iT^{2} \)
5 \( 1 + (-0.858 - 0.858i)T + 5iT^{2} \)
7 \( 1 + 1.49iT - 7T^{2} \)
13 \( 1 + (2.98 + 2.98i)T + 13iT^{2} \)
17 \( 1 - 2.42iT - 17T^{2} \)
19 \( 1 + (-3.38 + 3.38i)T - 19iT^{2} \)
23 \( 1 - 0.410T + 23T^{2} \)
29 \( 1 + (-6.31 - 6.31i)T + 29iT^{2} \)
31 \( 1 + 8.16iT - 31T^{2} \)
37 \( 1 + (1.37 + 1.37i)T + 37iT^{2} \)
41 \( 1 - 4.70T + 41T^{2} \)
43 \( 1 + (2.16 + 2.16i)T + 43iT^{2} \)
47 \( 1 + 0.963iT - 47T^{2} \)
53 \( 1 + (4.60 + 4.60i)T + 53iT^{2} \)
59 \( 1 + (2.45 - 2.45i)T - 59iT^{2} \)
61 \( 1 + (6.00 + 6.00i)T + 61iT^{2} \)
67 \( 1 + (5.95 + 5.95i)T + 67iT^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + (1.12 - 1.12i)T - 83iT^{2} \)
89 \( 1 + 5.40iT - 89T^{2} \)
97 \( 1 + 5.47T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.417375145312919644291596330407, −8.122575715793334647097989008315, −7.34326306979397728134349460819, −6.81740965927143738976214057651, −6.13749076232526724317714342374, −5.30324948692865176180499201993, −4.39882467543123900016345502736, −2.80190016713583341589768238086, −1.63211822588458073462057508044, −0.53000334853116799849327563819, 1.23655674376175068870725480984, 3.02974082397216127622858413795, 4.19246022726732132044923016226, 4.85904019443498829096670637155, 5.61887981505715231005061684783, 6.25167777008001072541230115854, 7.16280214423654508123292892273, 8.619667861809886766527937667015, 9.411194280938315305841536642510, 9.685386888168722862415385336614

Graph of the $Z$-function along the critical line